\documentclass{article}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{exam-zh-font}

\ExplSyntaxOn
\keys_set:nn { exam-zh }
  {
    math-font = asana ,  % yes
    % math-font = cambria ,  % yes
    % math-font = garamond ,  % yes
    % math-font = libertinus ,  % yes
    % math-font = lm ,   % yes
    % math-font = newcm ,  % yes
    % math-font = pala ,  % yes
    % math-font = stix ,  % yes
    % math-font = xits , % yes
    % math-font = none
  }
\ExplSyntaxOff

\fboxsep=0pt
\begin{document}

$e^{x_1}$
% $\fbox{$e$}^{x_1}$
$\cramped{\mathrm{e}^{x_1}}$
$\smash[t]{\mathrm{e}}^{x_1}$
$\smash{\mathrm{e}}^{x_1}$
$\cramped{\mathrm{e}^{x_1}}$
% $\fbox{$\mathrm{e}$}^{x_1}$

\end{document}